Emergence of Goldstone excitations in stress correlations of glass-forming colloidal dispersions

Abstract

We compute the dynamic correlations of the nonlocal shear stress at the liquid to glass transition in colloidal dispersions. Whereas Eshelby's elastic pattern is recovered independently of the dynamics, the Goldstone modes of the colloidal glass are diffusive, ω = −iDq 2 , because the transverse momentum is not conserved -excluding propagating transverse sound. Precursors of both, the Goldstone modes and the long-ranged, anisotropic stress correlations, can be observed in the colloidal fluid. The slow diffusive mode dominates the dynamics in the supercooled liquid for wave numbers qξ 1 with a correlation length ξ ∝ τ 1/2 which grows like the square root of the relaxation time τ . Time-and space-dependent stress correlations are anisotropic, decaying like r −5 . These results are derived within a hydrodynamic theory, generalising Maxwell's model to finite wave numbers, after the dynamics has been decomposed into potentially slow modes, associated with the order parameter, and fast microscopic degrees of freedom. Alternatively, hydrodynamic equations, closed by constitutive laws, can be used to predict the linear response of stress to an applied shear flow.

Stress correlation function and linear response of Brownian particles

“…in confined fluids [25]. Gratifyingly, our generalization leads to the same expression for the stress autocorrelation as in overdamped Newtonian and Langevin systems [5][6][7]. Furthermore, we will show in sect.…”

“…Our ansatz naturally leads to the complete coupling of the tensorial stress fluctuations to the vectorial particle displacements. In the end, we regain the formally identical expression for the stress autocorrelation and therefore for the linear response of the stress tensor as in systems where velocities are kept as dynamical variables [5,7]. In the final section, we argue that the irreducible memory kernel indeed is the correct one since it can be related to the Green-Kubo transport coefficients, viz.…”

“…Usually, such a system is described with the locations {r j } N and the momenta {p j } N of the particles. In [5], we used such a description to follow the system through the glass transition, and only after obtaining a general expression, the overdamped limit was considered. Here, we start with the overdamped simplification and consider N interacting particles performing Brownian motion.…”

“…In order to study the slow dynamics of the system resulting from the conserved hydrodynamic fields, a Zwanzig-Mori decomposition [12,13] of the dynamical variable of interest, the stress tensor σ, will be considered. Motivated by earlier works [5,7] and having the density as the only conserved dynamical variable, we choose the latter as the only distinguished variable. The Fourier-modes are given by…”

“…This question was already considered in [5]. There, a set of Langevin's equations of motion for the individual colloidal particles was investigated, which leads to a time evolution of the probability distribution function that is governed by the Klein-Kramers equation.…”